物理波中的拓扑指标公式：谱流、陈数与拓扑接触

Non-commutative odd Chern numbers and topological phases of disordered chiral systems

An odd index theorem for higher odd Chern characters of crossed product algebras is proved. It generalizes the Noether-Gohberg-Krein index theorem. Furthermore, a local formula for the associated cyclic cocycle is provided. When applied to the non-commutative Brillouin zone, this allows to define topological invariants for condensed matter phases from the chiral unitary (or AIII-symmetry) class in the presence of strong disorder and magnetic fields whenever the Fermi level lies in region of Anderson localization.

Chern numbers as half-signature of the spectral localizer

Two recent papers proved that complex index pairings can be calculated as the half-signature of a finite dimensional matrix, called the spectral localizer. This paper contains a new proof of this connection for even index pairings based on a spectral flow argument. It also provides a numerical study of the spectral gap and the half-signature of the spectral localizer for a typical two-dimensional disordered topological insulator in the regime of a mobility gap at the Fermi energy. This regime is not covered by the above mathematical results (which suppose a bulk gap), but nevertheless the half…

Intrinsic Chern-Connes Characters for Crossed Products by $\mathbb Z^d$

We present a natural imbedding of the crossed product $\mathcal A \rtimes_ξ\mathbb Z^d$ into the $C^\ast$-algebra of adjointable operators over the standard Hilbert $\mathcal A$-module $\mathcal H_{\mathcal A}$. By replacing the representations on Hilbert spaces with this canonical imbedding, we define Fredholm modules and corresponding Chern-Connes characters that are intrinsic to the $C^\ast$-dynamical system $(\mathcal A,ξ,\mathbb Z^d)$. The compression of the Dirac operator against projectors from $\mathcal A \rtimes_ξ\mathbb Z^d$ produces generalized Fredholm operators over $\mathcal H_{\…

量子谐振子、索引定理与谱不对称性

Quantum harmonic oscillator, index theorem and spectral asymmetry

研究量子谐振子的谱不对称性，揭示其配分函数与索引定理的联系，展现内能作为非SUSY的索引定理表现。

The edge spectrum of Chern insulators with rough boundaries

Chern insulators are periodic band insulators with the property that their projector onto the occupied bands have non-zero Chern number. Chern insulator with a homogeneous boundary display continuum spectrum that fills the entire insulating gap. The local density of states corresponding to this part of the spectrum is localized near the boundary, hence the terminology edge spectrum. An interesting question arises, namely, if a rough boundary, which can be seen as a strong random potential acting on these quasi 1-dimensional states, would destroy the continuum edge spectrum. This paper shows ho…

构造陈绝缘体的涡旋函数与基态：理想条件、来自指标定理的不等式以及冯·诺依曼格点上的类相干态

Constructing vortex functions and basis states of Chern insulators: ideal condition, inequality from index theorem, and coherent-like states on von Neumann lattice

本文在紧束缚框架下提出陈绝缘体涡旋函数与基态的构造理论，优化涡旋函数并推导拓扑不等式，构造两类基组，为分数陈绝缘体多体计算提供工具。

Chern特征标与费米点

Chern character and Fermi point

本文通过费米点提出拓扑K理论的Chern特征标，并证明了四维拓扑绝缘体的边缘指数偶数性和体-边对应关系。

A 3d-3d appetizer

We test the 3d-3d correspondence for theories that are labelled by Lens spaces. We find a full agreement between the index of the 3d ${\cal N}=2$ "Lens space theory" $T[L(p,1)]$ and the partition function of complex Chern-Simons theory on $L(p,1)$. In particular, for $p=1$, we show how the familiar $S^3$ partition function of Chern-Simons theory arises from the index of a free theory. For large $p$, we find that the index of $T[L(p,1)]$ becomes a constant independent of $p$. In addition, we study $T[L(p,1)]$ on the squashed three-sphere $S^3_b$. This enables us to see clearly, at the level of …

Index and Dynamics of Quantized Contact Transformations

Quantized contact transformations are Toeplitz operators over a contact manifold $(X,α)$ of the form $U_χ = ΠA χΠ$, where $Π: H^2(X) \to L^2(X)$ is a Szego projector, where $χ$ is a contact transformation and where $A$ is a pseudodifferential operator over $X$. They provide a flexible alternative to the Kahler quantization of symplectic maps, and encompass many of the examples in the physics literature, e.g. quantized cat maps and kicked rotors. The index problem is to determine $ind(U_χ)$ when the principal symbol is unitary, or equivalently to determine whether $A$ can be chosen so that $U_χ…

On the geometric quantization of contact manifolds

Suppose that $(M,E)$ is a compact contact manifold, and that a compact Lie group $G$ acts on $M$ transverse to the contact distribution $E$. In an earlier paper, we defined a $G$-transversally elliptic Dirac operator $\dirac$, constructed using a Hermitian metric $h$ and connection $\nabla$ on the symplectic vector bundle $E\rightarrow M$, whose equivariant index is well-defined as a generalized function on $G$, and gave a formula for its index. By analogy with the geometric quantization of symplectic manifolds, the $\mathbb{Z}_2$-graded Hilbert space $Q(M)=\ker \dirac \oplus \ker \dirac^{*}$ …

Moduli Spaces of Contact Instantons

We construct the moduli space of contact instantons, an analogue of Yang-Mills instantons defined for contact metric $5$-manifolds and initiate the study of their structure. In the $K$-contact case we give sufficient conditions for smoothness of the moduli space away from reducible connections and show the dimension is given by the index of an operator elliptic transverse to the Reeb foliation. The moduli spaces are shown to be Kähler when the $5$-manifold $M$ is Sasakian and hyperKähler when $M$ is transverse Calabi-Yau. We show how the transverse index can be computed in various cases, in pa…